The purpose of this study was to examine the differences in bully involvement of middle school students with learning disabilities and their peers without disabilities. Data were
examined to assess whether factors related to the bullying dynamic could be measured equivalently across the two subgroups of students. Once equivalence was established,
comparative analyses were conducted to determine differences in associations, latent means, and predictors related to the bullying dynamic. The following section will explicitly detail analyses conducted to address the four research questions. Specifically, this chapter will detail the sample demographics, imputation process, confirmatory factor analysis equivalence model, tests of associations related to bullying constructs, demographic predictors, and social supports predictors.
Sample Demographics
As reported in Chapter 3, Intraclass Correlation Coefficients (ICC) were not significant for school level outcome variables (i.e., bullying, victimization, fighting, anger), indicating that school level nesting variables are unnecessary for the current analyses. Therefore, data were analyzed at the respondent level for students with learning disabilities and students without disabilities. Self-reported demographic information was collected for gender, age, grade, race, grade point average (GPA), and participation in extracurricular activities. While this study is a cross-sectional investigation of differences between students with learning disabilities and students without disabilities, exploration of Waves 1 through 3 was necessary to recover some unreported demographic information due to respondents’ failure to report. Since it is reasonable to assume gender and race are static variables, and age and grade will increase as a function of
time, missing data for these variables were inserted as necessary. GPA and participation in extracurricular activities are not fixed variables, so missing data were imputed using a multiple imputation procedure, which is described in the following section.
To investigate sampling differences between students with learning disabilities and students without disabilities, 2 statistics were calculated. Overall, there was not a significant difference between the groups for school, age, grade, race, GPA, and extracurricular
participation, indicating the samples are proportionally similar (see Table 13). However, the 2 statistic revealed significant differences for gender (2 (1)= 5.18, p < .05). Descriptive statistics (see Table 14) revealed that gender was relatively proportional for students without disabilities (53.6% female, 46.4% male), but males were overrepresented in the group of students with learning disabilities (39.8% female, 60.2% male).
Table 13 2
Difference Tests for Variables Across Disability Type
Demographic Variables 2 df p School 1.87 3 .600 Gender 5.18 1 .023* Age 5.35 4 .254 Grade 0.28 1 .599 Race 1.63 5 .897 GPA 7.94 7 .326 Extracurricular 0.05 1 .824
* represents significant at the .05 level ** represents significant at the .001 level
Table 14
Descriptive Statistics for Students From the Two Groups
No Disability (%) Learning Disability (%)
Demographics N = 360 N = 83 School School 1 91 (25.3) 18 (21.7) School 2 98 (27.2) 21 (25.3) School 3 68 (18.9) 14 (16.9) School 4 103 (28.6) 30 (36.1) Gender Female 193 (53.6) 33 (39.8) Male 167 (46.4) 50 (60.2) Age 11 3 (0.8) -- 12 118 (32.8) 21 (25.3) 13 159 (44.2) 35 (42.2) 14 74 (20.6) 24 (28.9) 15 6 (1.7) 3 (3.6) Grade 7 172 (47.8) 37 (44.6) 8 188 (52.2) 46 (55.4) Race
American Indian or Alaska Native 8 (2.2) 2 (2.4)
African American 193 (53.6) 48 (57.8)
Asian 4 (1.1) 2 (2.4)
Hispanic 21 (5.8) 4 (4.8)
White 97 (26.9) 20 (24.1)
Other 37 (10.3) 7 (8.4)
Grade Point Average Missing: 11 (3.1) Missing: 2 (2.4)
Mostly A’s 53 (14.7) 10 (12.0)
Mostly A’s & B’s 161 (44.7) 27 (32.5)
Mostly B’s 15 (4.2) 6 (7.2) Mostly B’s & C’s 57 (15.8) 16 (19.3) Mostly C’s 21 (5.8) 8 (9.6) Mostly C’s & D’s 16 (4.4) 4 (4.8) Mostly D’s & F’s 5 (1.4) 2 (2.4) Not Sure 21 (5.8) 8 (9.6) (continued)
Table 14 (continued)
No Disability (%) Learning Disability (%)
Demographics N = 360 N = 83
Participation in Extracurricular
Activities Missing: 18 (5.0) Missing: 3 (3.6)
No 142 (39.4) 33 (39.8)
Yes 200 (55.6) 47 (56.6)
Percentage of Services Missing: 2 (2.4)
No Services Received 360 (100) --
20% or Less -- 30 (36.1)
21 – 60% -- 24 (28.9)
61% or More -- 27 (32.5)
Descriptive statistics by gender and disability. To further investigate sampling
differences, separate cross tabulations with corresponding 2 statistics were calculated for gender by disability, race by disability, females by gender and disability, and males by gender and disability. Specific demographic cross tab data are reported in Appendix E, and separate 2 statistics are reported in Table 15. The 2 statistics for gender by disability uncovered
nonsignificant results for school, age, grade, race, participation in extracurricular activities, and percentage of special education services. The 2 for grade point average of students without disabilities (2 (7)= 25.10, p < .001) revealed that females report higher GPAs (67.4% with at least A’s and B’s) than males (50.3 with at least A’s and B’s).
Descriptive statistics by race and disability. Similar to gender, 2 statistics for race by disability uncovered nonsignificant results for gender, age, grade, GPA, participation in
extracurricular activities, and percentage of special education services. However, 2 for school of students without disabilities (2 (15)= 115.17, p < .001) revealed that school 1 and 2 represented 91.8% of the white students without disabilities, and schools 3 and 4 represented 67.4% of the African American students without disabilities.
Descriptive Statistics by race, gender, and disability. This variation is also evident in the 2 statistics for Females by race and disability (2 (15)= 62.10, p < .001), males by race and disability (2 (15)= 63.91, p < .001) for students without disabilities, and males by race and disability (2 (15)= 32.29, p < .01) for students with disabilities. More specifically, school 4 represented 57.6% of the African American males with learning disabilities. However, school level analyses were not conducted because Intraclass Correlations Coefficients revealed a nonsignificant difference among the schools. Finally, 2 for Males and Females by race and disability label revealed nonsignificant results for age, grade, GPA, participation in
extracurricular activities, and percentage of special education services. Overall, the 2 statistics reported in Table 15 reveal that the current sample is relatively similar across the selected demographic items.
Table 15 2
Statistics for Cross Tabulations
Demographic 2 df P
2 by Gender and no Disability
School 3.28 3 .350 Age 9.39 4 .052 Grade 3.02 1 .082 Race 3.09 5 .686 GPA 25.10 7 .001** Extracurricular .71 1 .413
2 by gender and learning disability
School 3.77 3 .288 Age .98 3 .805 Grade .10 1 .748 Race 4.81 5 .439 GPA 7.13 7 .435 Extracurricular .01 1 .622 Percentage of Services 1.19 2 .516 (continued)
Table 15 (continued)
Demographic 2 df P
2 by race and no disability
School 115.17 15 .000** Gender 3.09 5 .686 Age 25.26 20 .192 Grade 3.03 5 .695 GPA 34.91 35 .473 Extracurricular 9.90 5 .078
2 by race and learning disability
School 24.38 15 .059 Gender 4.81 5 .439 Age 9.47 15 .851 Grade 3.00 5 .700 GPA 47.67 35 .075 Extracurricular 5.67 5 .339 Percentage of Services 8.631 10 .567
2 by race, female, and no disability
School 62.10 15 .000**
Age 18.61 15 .232
Grade 8.22 5 .145
GPA 25.08 35 .892
Extracurricular 5.41 5 .368
2 by race, male, and no disability
School 63.91 15 .000** Age 19.04 20 .519 Grade 2.23 5 .816 GPA 38.07 35 .331 Extracurricular 9.91 5 .078 (continued)
Table 15 (continued)
Demographic 2 df P
2 by race, female, and learning disability
School 12.32 15 .654 Age 9.62 15 .843 Grade 3.00 5 .700 GPA 28.43 35 .776 Extracurricular 3.92 4 .561 Percentage of Services 8.70 10 .561
2 by race, male, and learning disability
School 32.29 15 .006* Age 15.74 15 .399 Grade 5.43 5 .365 GPA 48.28 35 .067 Extracurricular 8.84 5 .116 Percentage of Services 14.78 10 .140 ** p < .001 * p < .01
Missing Data Procedures
Data imputation. To address the issue of missing data within the current sample, a missing data pattern analysis was conducted in SPSS 18.0 (PSAW, 2009) and a multiple
imputation procedure was executed using the PROC MI function in SAS 9.2 (SAS, 2008). Since missingness can bias a sample (Davey, Savla, & Luo, 2005; Rubin, 1976), it was necessary to account for the missing values to best represent students with learning disabilities and students without disabilities. While Little’s MCAR (Little, 1988) test was insignificant (2 (1136)= 1150.03,
p = .379), the sample may not necessary be characterized as Missing Completely at Random
because the missingness on any given variable may be related to the observed data (Enders & Peugh, 2004; Luengo, García, & Herrera, 2010). Therefore, the data for the current sample will be approached as Missing at Random (MAR), because the missingness on any given variable is
not related to itself, but it may be related to another measured variable (Enders & Peugh, 2004; Luengo et al., 2010). However, MAR is an assumption and according to Schafer and Graham (2002), “there is no way to test whether MAR holds in a data set” (p 152) because data cannot be directly obtained from nonresponders. Collins, Shafer, and Kam (2001) demonstrated that the MAR assumption has minimal impact on estimates and standard errors.
Overall, 36 (81.8%) out of the 44 measured variables included some missing data, with 71 (16.0%) out of the 443 respondents having some level of missingness. However, the
missingness of the total sample was extremely low, with only 1.7% (332) missing from the measured items by total number of respondents (see Figure 6). Overall, missingness per item ranged from 0 to 4.7%. A missingness breakdown of the items is reported in Table 16.
Table 16
Missing Data Specifics per Item
Item Total Missing Percent Missing
Participation in Extracurricular Activities 21 4.74 I am treated with as much respect as other students
are.
14 3.16 What is your overall grade average this year? 13 2.93
There is at least one teacher or other adult in this school I can talk to if I have a problem.
12 2.71
I feel proud of belonging to ___ Middle School 12 2.71 I have friends who help me with practical problems… 11 2.48 I have friends I can talk to, who give good suggestions
and advice about my problems.
11 2.48
I lost my temper for no reason. 10 2.26
I threatened to hurt or hit another student. 10 2.26 I fought other students I could easily beat. 10 2.26
I was angry all day. 9 2.03
I encouraged people to fight. 9 2.03
Other students called me “gay.” 9 2.03
I spread rumors about other students. 9 2.03
I got in a physical fight. 9 2.03
In a group I teased other students. 9 2.03
There are people in my family who help me with practical problems…
9 2.03
At school, there are adults who help me with practical problems…
9 2.03 (continued)
Table 16 (continued)
Item Total Missing Percent Missing
I have friends I can talk to… 9 2.03
As school, there are adults I can talk to… 9 2.03
The teachers here respect me. 9 2.03
I got hit and pushed by other students. 8 1.81
I was mean to someone when I was angry. 8 1.81
I teased other students. 8 1.81
I hit back when someone hit me first. 8 1.81
I started (instigated) arguments or conflicts. 8 1.81
Other students picked on me. 8 1.81
There are people in my family I can talk to, who give
good suggestions and advice about my problems. 8 1.81
As school, there are adults I can talk to, who give
good suggestions and advice about my problems. 8 1.81
There are people in my family I can talk to… 8 1.81
I called other students “gay.” 7 1.58
Other students called me names. 7 1.58
I got in a physical fight because I was angry. 7 1.58
I helped harass other students. 7 1.58
I upset other students for the fun of it. 7 1.58
Percentage of Special Education Services 2 0.05
Although Luengo and colleagues (2010) suggest that missing data between 1 and 5% are generally manageable, a multiple imputation procedure was employed to preserve the integrity of each group of respondents and create a parsimonious dataset. Using Kärnä and colleagues (in press) as a model, data were imputed with the SAS PROC MI function, using the MCMC algorithm. In total, 100 imputations were conducted separately for students with learning disabilities and students without disabilities. Next, the average imputed value for each missing data point was calculated, which according to Kärnä and colleagues (in press) “represents the best population estimate of the value need to reproduce the population parameters” (p. 55). Overall, one parsimonious data set was created, which best represents the sample population.
Factor parceling. Following the multiple imputation process, an item-to-construct balancing procedure was conducted to create parcels for bullying, victimization, fighting, and sense of belonging (Little et al., 2002). Parcels, which are aggregate-level indicators, were created to establish a just-identified measurement model because the focus of this study hinges on constructs, not item-level indicators. Additionally, “a just-identified construct has only one unique solution that optimally captures the relation among the items, no matter what other constructs are considered or included in a model” (Little et al., 2002, p. 162). Since three individual indicators theoretically define the anger and sense of belonging scales, the parceling procedure was unnecessary because they are already just-identified. Therefore, separate single- construct models were created for bullying, victimization, fighting, and sense of belonging using maximum likelihood estimation. Once the models were created, the three highest loadings were used to anchor the construct, and the next highest loadings were added to the anchors in inverse order (Little et al., 2002). Due to the item total for victimization, fighting, and sense of
items with the lowest loadings were combined and averaged to create the third parcel. Table 17 contains the final constructed parcels.
Table 17
Item Parceling Procedure for the Eight Subscales
Parcel #
Items Factor Loadings
Bullying
1 Ques23A: I upset other students for the fun of it. Ques23O: I encouraged people to fight.
Ques23H: I helped harass other students.
.876
2 Ques23B: In a group I teased other students.
Ques23G: I started (instigated) arguments or conflicts. Ques23K: I threatened to hurt or hit another student.
.832
3 Ques23P: I teased other students.
Ques23F: I spread rumors about other students.
.773
Victimization
1 Ques23N: Other students called me names. .901
2 Ques23D: Other students picked on me. .798
3 Ques23T: I got hit and pushed by other students. Ques23J: Other students called me “gay.”
.713
Fighting
1 Ques23E: I got in a physical fight .846
2 Ques23L: I got in a physical fight because I was angry. .702 3 Ques23I: I hit back when someone hit me first.
Ques23C: I fought other students I could easily beat.
.612
Anger
1 Ques23S: I was angry all day. .729
2 Ques23R: I was mean to someone when I was angry. .709
3 Ques23M: I lost my temper for no reason. .683
Table 17 (continued)
Parcel # Item Factor
Loading Sense of Belonging
1 Ques14B: I am treated with as much respect as other students are. .737
2 Ques14C: The teachers here respect me. .705
3 Ques14A: I feel proud of belonging to __ Middle School.
Ques14D: There is at least one teacher or other adult in this school I can talk to if I have a problem.
.605
Support: School
1 Ques17D: At school, there are adults I can talk to, who give good suggestions and advice about my problems.
.926 2 Ques17A: At school, there are adult I can talk to, who care about my
feelings and what happens to me. .764
3 Ques17G: At school, there are adults who help me with practical problems…
.655
Social Support: Family
1 Ques17E: There are people in my family I can talk to, who give me good suggestions and advice about my problems.
.837 2 Ques17H: There are people in my family who help me with practical
problems…
.810 3 Ques17B: There are people in my family I can talk to, who care
about my feeling and what happens to me.
.720
Social Support: Peers
1 I have friends I can talk to, who give good suggestions and advice about my problems.
.811 2 I have friends I can talk to, who care about my feelings and what
happens to me.
.801 3 I have friends who help me with practical problems… .760
Once the parcels were established for the eight separate constructs, data were aggregated across the 100 imputations to represent the best population estimate for the imputed data (Kärnä et al., in press) using the Aggregate function in SPSS 18.0 (PSAW, 2009). This aggregation provided a single “super matrix” based on the mean scores of the 100 imputations for each respondent on the eight constructs. This aggregated dataset was used for all consequent data
analytic procedures, and mean scores and standard deviations of each parcel for students without disabilities and students with learning disabilities are reported in Table 18.
Table 18
Means and Standard Deviations by Subgroup for Individual Parcels
Students without Disabilities Students with Learning Disabilities
Parcel Mean sd Mean sd
Bully 1 1.35 .57 1.31 .45 2 1.44 .59 1.39 .51 3 1.31 .56 1.26 .48 Victimization 1 1.58 1.07 1.53 .98 2 1.62 1.09 1.48 .97 3 1.36 .68 1.38 .67 Fighting 1 1.52 .99 1.47 .84 2 1.35 .82 1.45 .93 3 1.64 .81 1.94 .92 Anger 1 1.56 1.02 1.48 1.01 2 1.52 .92 1.42 .66 3 1.37 .83 1.31 .60 Sense of Belonging 1 2.87 .78 2.94 .70 2 3.06 .76 3.04 .76 3 3.07 .66 3.12 .65 Support: School 1 2.11 .59 2.27 .66 2 2.15 .63 2.22 .70 3 2.23 .63 2.18 .64 (continued)
Table 18 (continued)
Students without Disabilities Students with Learning Disabilities
Parcel Mean sd Mean sd
Social Support: Family
1 2.56 .59 2.51 .65
2 2.54 .59 2.38 .65
3 2.71 .48 2.59 .62
Social Support: Peers
1 2.23 .61 2.21 .67
2 2.29 .64 2.24 .71
3 2.25 .60 2.25 .62
Construct Equivalence
Can the constructs that define the bullying dynamic be measured equivalently across students with learning disabilities and students without disabilities? This research question addressed measurement invariance on the University of Illinois Aggression Scales (Espelage & Holt, 2001), Sense of Belonging Scale (Espelage & Holt, 2001) and Social Support Record (Vaux, 1988). To evaluate this question, a multi-group confirmatory factor analysis procedure was utilized. This stepwise process enables the measurement equivalences of
constructs and allows for direct comparisons among groups (Little, 1997; Shogren et. al., 2007). To establish measurement invariance and discern that students with learning disabilities and students without disabilities are interpreting the constructs equivalently, strong (e.g., intercept) invariance must be established (Little, 1997). This process includes three distinct steps: (a) test the model fit based on manifest indicators, (b) equate factor loadings across groups and evaluate model fit, and (c) equate intercepts across groups and evaluate model fit.
Overall, eight latent constructs were used in the present measurement model to test measurement equivalence between students with learning disabilities and students without
disabilities. These constructs include: (a) bullying, (b) victimization, (c) fighting, (d) anger, (e) sense of belonging, (f) school support, (g) family social support, and (h) peer social support. As stated in the data imputation section of this chapter, three parcels or item level indicators were used to create each construct to maintain a just-identified model.
Using Shogren and colleagues’ (2007) CFA procedure as a model, an indicator loading procedure (i.e., effects coding) was used by constraining the sum of the indicator’s loadings to the total number of indicators (e.g. LY(1,1) = 3 – LY(2,1) – LY(3,1)). While traditional techniques are generally used for CFA procedures, the effects coding method allows for the estimation of a construct’s latent variance in a non-arbitrary metric (Little, Slegers, & Card, 2006; Shogren et al., 2007). To maintain consistency, intercepts were estimated using a similar procedure (CO TY(1) = 0 – TY(2) – TY(3)).
As a preliminary step, separate models were fit for each group of students to determine if they if the initial parameters were tenable for each group of students. The freely estimated model for students without disabilities demonstrated acceptable model fit (2(224) = 558.06, p <.001, RMSEA = .061, NNFI = .95, CFI = .096), and the freely estimated model for students with learning disabilities demonstrated acceptable model fit on RMSEA and mediocre fit for NNFI and CFI (2(224) = 384.28, p <.001, RMSEA = .071, NNFI = .86, CFI = .88). Since both freely estimated models fell within the acceptable range on at least one of the fit indices, it was appropriate to move forward with the CFA.
Using the effects coding method described earlier, a multiple group confirmatory analysis was conducted with two groups. Since parcels were used to estimate the constructs, a just-
identified model was established for both groups with 24 parcels estimating 8 separate
freely estimated. The configural model demonstrated an acceptable fit based on the relative fit indices (2(448)=942.344, p < .001, RMSEA = .063, NNFI = .92, CFI = .94). The initial step of the CFA indicates that with the same measurement model compared across the two groups, the overall fit was acceptable.
Following the configural invariance step, factor loadings are equated to determine if they are invariant between the two groups of students. The loading invariance test revealed that the model remained within the acceptable range for the appropriate fit indices (2(464) = 942.344, p < .001, RMSEA = .063, NNFI = .92, CFI = .94). The RMSEA model test was conducted to
establish if the constraints are tenable, where the RMSEA value of the constrained model is examined to determine if it falls within the 90% confidence interval of the freely estimated model (Little, 1997). Additionally, changes in the Comparative Fit Index (CFI) of less than .01 indicate that the constraints are tenable (Cheung & Rensvold, 2002). Based on the RMSEA model test and evaluation of the CFI, it was concluded that the loadings are invariant between students with learning disabilities and students without disabilities.
After establishing loading invariance, constraints were placed on the intercepts to determine if they were invariant between groups. As with the previous two models (i.e. freely estimated, loading invariant) the strong metric invariance model maintained acceptable model fit (2(480) = 1007.933, RMSEA = .063, NNFI = .92, CFI = .94). Additionally, the strong metric model met the criteria of the RMSEA model test and CFI evaluation, indicating that no significant changes were documented in model fit as constraints increased. Based on these statistics; bullying, victimization, fighting, anger, sense of belonging, and social supports (i.e., Teacher, Family, Peer) are being equivalently assessed for students with learning disabilities and
students without disabilities. Table 19 contains loadings, intercepts, and estimated latent variances from the strong metric invariance model.
Table 19
Loadings, Intercepts, and Estimated Latent Variance From Strong Metric Invariance Model
Indicator - Loading
Estimates (SE) Estimates (SE) - Intercept - Standardized Loadingsa Bully Parcel 1 .98 (.03) .02 (.04) .82 Parcel 2 1.11 (.03) -.08 (.04) .89 Parcel 3 .91 (.03) .07 (.04) .77 Victimization Parcel 1 1.22 (.04) -.27 (.06) .89 Parcel 2 1.12 (.04) -.10 (.06) .80 Parcel 3 .66 (.03) .37 (.05) .74 Fighting Parcel 1 1.16 (.05) -.26 (.08) .77 Parcel 2 .97 (.05) -.10 (.07) .73 Parcel 3 .87 (.05) .35 (.07) .66 Anger Parcel 1 1.12 (.05) -.09 (.08) .70 Parcel 2 1.07 (.05) -.07 (.07) .77 Parcel 3 .81 (.05) .16 (.07) .65 Belonging Parcel 1 1.04 (.06) -.25 (.17) .68 Parcel 2 1.04 (.06) -.07 (.17) .68 Parcel 3 .92 (.05) .32 (.16) .69 Support: School Parcel 1 1.09 (.03) -.23 (.08) .88 Parcel 2 1.04 (.04) -.09 (.08) .79 Parcel 3 .88 (.04) .31 (.09) .68
Table 19 (continued) Indicator - Loading Estimates (SE) - Intercept Estimates (SE) - Standardized Loadingsa Social Support: Family
Parcel 1 1.11 (.04) -.32 (.10) .84
Parcel 2 1.10 (.04) -.32 (.10) .82
Parcel 3 .80 (.04) .64 (.09) .70
Social Support: Peers
Parcel 1 1.03 (.03) -.11 (.08) .82
Parcel 2 1.03 (.04) -.03 (.08) .79
Parcel 3 .94 (.04) .14 (.08) .77
aCommon Metric Completely Standardized Solution
The overall model fit statistics for the three step, multi-group confirmatory factor analysis are presented in Table 20, with Figure 7 representing the strong metric invariance model. In addition to establishing strong metric invariance between the groups and across the eight latent constructs; latent means, unique residuals, and squared multiple correlations were calculated when constraining the loadings and intercepts. These statistics are reported in Table 21. Given the results of the multi-group CFA, measurement invariance has been established, and research question one confirmed.
Table 20
Fit Indices for Multi-Group Confirmatory Factor Analysis
Model 2 Df P RMSEA RMSEA
90% CI NNFI CFI Constraint Tenable Configural Invariance 942.344 448 <.001 .063 .056 - .070 .92 .94 -- Loading Invariance 979.358 464 <.001 .063 .057 - .070 .92 .94 Yes Intercept Invariance 1007.933 480 <.001 .063 .056 - .069 .92 .93 Yes
Table 21
Mean Scores, Unique Residuals, and Squared Multiple Correlations for Individual Parcels